Low complexity maximum likelihood sequence detection system and methods

ABSTRACT

A system and method implement low complexity maximum likelihood sequence detection. A decision feedback algorithm computes x(M+D+L−1). Optimality examination is performed for x(M), and state values and values of Markov states along paths from states in x(M) to x1(M+L) are computed.

RELATED APPLICATIONS

This application claims priority to U.S. Provisional Application Ser. No. 60/854,867, filed Oct. 25, 2006 and titled “Low Complexity Maximum Likelihood Sequence Detection Method”, incorporated herein by reference.

BACKGROUND

Consider the scenario where a sequence of vector symbols, with each vector having K binary elements, are sent from a transmitter to a receiver through a vector intersymbol interference (ISI) channel, whose number of taps is L, subject to additive Gaussian noise. Assume the source vector symbols are independently generated with all possible values being equal probable. If the receiver is willing to minimize the probability of sequence detection error, the optimal decision is given by the maximum likelihood (ML) sequence that maximizes the log likelihood function. Finding such sequence is known as the maximum likelihood sequence detection (MLSD) problem.

Conventionally, the ML sequence is computed using the well known Viterbi algorithm (VA), whose complexity scales linearly in the sequence length, but exponentially in the source symbol vector length K, and exponentially in the number of ISI channel taps L. Such complexity can be prohibitive for systems with large KL values. Throughout the past three decades, many attempts have been made to find sequence detectors performing about the same as the VA, but less complex in terms of the scaling law in the Markov states. The main idea considered in these algorithms is to update only a selected number of routes upon the reception of each observation so that the worst case complexity of the algorithm is under control. However, a consequence of such limited search is that none of these complexity-reduction methods can guarantee the ML sequence, which is the sequence that maximizes the log likelihood function. On the other hand, if the length of the input vector sequence, N, is small, one can regard the MLSD problem as a maximum likelihood (ML) lattice decoding problem with an input symbol vector of length NK.

Consequently, ML sequence can be obtained using various versions of the sphere decoding algorithm with low average complexity, under the assumption of high signal to noise ratio (SNR). Unfortunately, due to the difficulty of handling a lattice of infinite dimension, these algorithms cannot extend directly to the situation of stream input where the length of the source sequence is practically infinity. In summary, most existing complexity reduction methods for MLSD either cannot guarantee the ML sequence, or are not suitable for stream input.

SUMMARY OF THE INVENTION

Although the VA is computation efficient in the sense of exploiting the underlying Markov chain structure, it does not fully exploit the statistical information of the system. Particularly, the observations of the system are related to the Markov states through a statistic model, which is usually known to the receiver. If the observation perturbation is small, the observation sequence provides a strong inference about the underlying Markov states. Such information can be used to significantly reduce the number of routes one should visit in the VA. For the communication system of this disclosure, an examination method is provided which guarantees the truncated sequence passing the examination is indeed the truncated ML sequence. As SNR goes to infinity, the examination method becomes highly efficient in the sense of passing the actual truncated source sequence with asymptotic probability one. Together with the help of an asymptotically efficient sequential detector whose probability of symbol detection error is asymptotically zero, the algorithm of the method obtains the ML sequence with an asymptotic complexity of O(LK²) per symbol. This complexity is, in the scaling order sense, no higher than any of the efficient sequence detectors, including suboptimal ones, that can achieve diminishing symbol detection error as SNR goes to infinity. In the situation of finite-length input sequence, the worst case complexity of the MLSD algorithm is in the same order of the VA.

The new MLSD algorithm of this disclosure is presented in a simple form in order to show clearly the insight of asymptotic complexity reduction. We make no effort in reducing the complexity further as long as the desired asymptotic scaling law is achieved. The proofs of the theorems are presented in a paper titled “On Low Complexity Maximum Likelihood Sequence Detection under High SNR” of U.S. Provisional Application Ser. No. 60/854,867, filed Oct. 25, 2006, incorporated herein by reference.

In one embodiment, a method implements low complexity maximum likelihood sequence detection, including the steps of: computing x(M+D+L−1) using a decision feedback algorithm; performing optimality examination for x(M); and computing state values and values of Markov states along paths from states in x(M) to x_(l)(M+L).

In another embodiment, a system implements low complexity maximum likelihood sequence detection. A detector captures an input signal. An optimality examiner processes the input signal to determine its optimality, and an advanced decoder analyzes the input signal further if the signal is not determined optimal.

In another embodiment, a method implements low complexity maximum likelihood sequence detection, including the steps of: initializing M=0; using a decision feedback detector to obtain a decision sequence {{circumflex over (x)}(n)≦D+L−2}; computing {{circumflex over (x)}(M+L−1)} using the decision feedback detector; using an optimality examination method to check whether {circumflex over (x)}(M)=x^(ML)(M), based upon {{circumflex over (x)}(n)|M−D−L+1<n<M+D+L−1}; putting {circumflex over (x)}(M) in the search list for time M if {circumflex over (x)}(M)=x^(ML)(M), and putting possible input symbols in the search list for time M if {circumflex over (x)}(M)≠x^(ML)(M); finding ML sequence using a revised Viterbi algorithm to search symbols in the search list; incrementing M; and repeating the steps of computing, using an optimality examination method, putting 1 to three for M.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 shows one exemplary system embodiment for low complexity maximum likelihood sequence detection.

FIG. 2 shows assumptions used to model the system of FIG. 1.

FIG. 3 shows a first theorem for an optimality examination method to verify whether a particular vector in a finite-length truncated decision sequence is identical to the corresponding vector in the ML sequence, in an embodiment.

FIG. 4 shows one exemplary maximum likelihood sequence detection (MLSD) algorithm, in an embodiment.

FIG. 5 shows a second theorem for an optimality examination method to verify whether a particular vector in a finite-length truncated decision sequence is identical to the corresponding vector in the ML sequence, in an embodiment.

FIG. 6 is a flowchart illustrating one exemplary method for maximum likelihood sequence detection, in an embodiment.

FIG. 7 is a flowchart illustrating one exemplary method for maximum likelihood sequence detection, in an embodiment.

DETAILED DESCRIPTION OF THE FIGURES

FIG. 1 shows one exemplary system for low complexity maximum likelihood sequence detection. An input signal 102 is captured by a detector 104 as signal 106 and passed to an optimality examiner 108. Optimality examiner 108 processes signal 106 to determine its optimality and produces output signal 110 if optimal and intermediate signal 112 if not optimal. Where optimality of signal 106 is not optimal, advanced decoder 114 is used to further analyze signal 106, passed as non-optimal signal 112, to produce output signal 116.

System Model

FIG. 2 shows assumptions 200 used to model system 100, FIG. 1.

Sequence Detection

FIG. 3 shows a first theorem 300 for an optimality examination method to verify whether a particular vector in a finite-length truncated decision sequence is identical to the corresponding vector in the ML sequence.

Given the observation sequence {y(n)} and a decision sequence

$\left\{ {\hat{x}(n)} \right\},{{{let}\mspace{14mu} 0} < \delta < {\frac{\lambda_{0}}{L}.}}$

Let D be a positive integer satisfying

$D > {{\frac{1}{\delta}\left( {{2\; U_{0}} + {\frac{L - 1}{L}\lambda_{0}}} \right)} - {\frac{3}{2}L} + 2.}$

For any 0≦M<N, if for all M−D<n<M+D+L−1, we have

${{{y(n)} - {\sum\limits_{l = 0}^{L - 1}\; {{F\lbrack l\rbrack}{\hat{x}\left( {n - l} \right)}}}}}^{2} < {\frac{\lambda_{0}}{L} - {\delta.}}$

Then {circumflex over (x)}(M)=x^(ML)(M) must be true.

The Simple MLSD Algorithm

FIG. 4 shows one exemplary maximum likelihood sequence detection (MLSD) algorithm 400.

System Model

FIG. 5 shows a second theorem 500 for an optimality examination method to verify whether a particular vector in a finite-length truncated decision sequence is identical to the corresponding vector in the ML sequence.

FIG. 6 is a flowchart illustrating one exemplary method 600 for maximum likelihood sequence detection. In step 602, method 600 computes x(M+D+L−1) using decision feedback algorithm (12) shown in FIG. 4. In step 604, method 600 perform optimality examination for x(M). In one example of step 604, first theorem 300, FIG. 3, is utilized to perform optimality examination for x(M). In step 606, method 600 computes state values and values of Markov states along paths from states in x(M) to x_(l)(M+L).

FIG. 7 is a flowchart illustrating one exemplary method 700 for maximum likelihood sequence detection. In step 702, method 700 initializes M to zero. In step 704, method 700 uses a decision feedback detector to obtain decision sequence {{circumflex over (x)}(n)|n≦D+L−2}. In step 706, method 700 uses the decision feedback detector to compute {{circumflex over (x)}(M+L−1)}. In step 708, method 700 checks whether {circumflex over (x)}(M)=x^(ML)(M) using the optimality examination method, based on {{circumflex over (x)}(n)|M−D−L+1<n<M+D+L−1}. In one example of step 708, first theorem 300, FIG. 3, is used as a basis for the optimality examination method. In step 710, method 700 puts {circumflex over (x)}(M) in a search list for time M, if {circumflex over (x)}(M) passed the examination and puts possible input symbols in the search list for time M, if {circumflex over (x)}(M) did not pass. In step 712, method 700 finds ML sequence using a revised Viterbi algorithm to search the symbols in the search list. In step 714, method increments M. As shown, Steps 706-714 repeat for each M.

In considering the maximum likelihood sequence detection (MLSD) problem of transmitting a sequence of binary vector symbols over a vector intersymbol interference channel, it is shown that as the signal to noise ratio (SNR) goes to infinity, the ML sequence can be obtained with a complexity of O(LK²) per symbol, where L is the number of channel taps and K is the vector length of the source symbol, under certain conditions. Such a complexity is no higher in order than any of the efficient sequence detectors, including suboptimal ones, that may achieve diminishing symbol detection error as SNR goes to infinity.

Changes may be made in the above methods and systems without departing from the scope hereof. It should thus be noted that the matter contained in the above description or shown in the accompanying drawings should be interpreted as illustrative and not in a limiting sense. The following claims are intended to cover all generic and specific features described herein, as well as all statements of the scope of the present method and system, which, as a matter of language, might be said to fall there between. 

1. A method for low complexity maximum likelihood sequence detection, comprising: computing x(M+D+L−1) using a decision feedback algorithm; performing optimality examination for x(M); and computing state values and values of Markov states along paths from states in x(M) to x_(l)(M+L).
 2. A system for low complexity maximum likelihood sequence detection, comprising: a detector for capturing an input signal; an optimality examiner for processing the input signal to determine its optimality; and an advanced decoder for farther analyzing the input signal if it is not determined optimal.
 3. A method for low complexity maximum likelihood sequence detection, comprising: initializing M=0; using a decision feedback detector to obtain a decision sequence {{circumflex over (x)}(n)|n≦D+L−2}; computing {{circumflex over (x)}(M+L−1)} using the decision feedback detector; using an optimality examination method to check whether {circumflex over (x)}(M)=x^(ML)(M), based upon {{circumflex over (x)}(n)|M−D−L+1<n<M+D+L−1}; putting {circumflex over (x)}(M) in the search list for time M if {circumflex over (x)}(M)=x^(ML)(M), and putting possible input symbols in the search list for time M if {circumflex over (x)}(M)≠x^(ML)(M); finding ML sequence using a revised Viterbi algorithm to search symbols in the search list; incrementing M;, and repeating the steps of computing, using an optimality examination method, putting 1 to three for M. 